Continuous Adjoint Method for Unstructured Grids

Abstract

Adjoint-based shape optimization methods have proven to be computationally efficient for aerodynamic problems. The majority of the studies on adjoint methods have used structured grids to discretize the computational domain. Because of the potential advantages of unstructured grids for complex configurations, in this study we have developed and validated a continuous adjoint formulation for unstructured grids. The hurdles posed in the computation of the gradient for unstructured grids are resolved by using a reduced gradient formulation. The methods to impose thickness constraints on unstructured grids are also discussed. The results for two- and three-dimensional simulations of airfoils and wings in inviscid transonic flow are used to validate the design procedure. Finally, the design procedure is applied to redesign the shape of a transonic business jet configuration; we were able to reduce the inviscid drag of the aircraft from 235 to 216 counts resulting in a shock-free wing. Although the Euler equations are the focus of the study in this paper of the adjoint-based approach, the solution of the adjoint system and gradient formulation can be conceptually extended to viscous flows. The approach presented in this study has been successfully used by the first and third authors for viscous flows using structured grids. However, particular aspects of the design process, such as the robustness of the mesh deformation process for unstructured grids, need more attention for viscous flows and are therefore the subject of ongoing research.